Title | **Noncommutative Gröbner Basis over a Divisible and Annihilable Ring** |

Author(s) | Nafissatou Diarra, Djiby Sow |

Type | Article in Conference Proceedings |

Abstract | The main objective of this article is to study noncommutative Gröbner bases over a divisible and annihilable ring (D-A ring). Such rings were introduced by Deepak Kapur and Yongyang Cai, and an algorithm for computing Gröbner bases in the commutative case was also given. If $I$ is an ideal of the associative algebra $V<x_1,...,x_n>=V<X>$ with non-commuting variables $x_1,...,x_n$ over a valuation ring $V$, a method for computing a Gröbner basis of $I$ was proposed recently. This method solves the membership problem in $I$ but does not allow to compute in the quotient ring $V<X>/I$. We generalized the method of Kapur and Cai in the noncommutative case. Our method allows to compute in the quotient ring $D<X>/I$, where $D$ is a D-A ring. This new approach for Gröbner basis over a D-A ring can have some applications in cryptography such as the study of the public key cryptosystem NTRU in ZniX hX2 1i where Zni is a D-A rin |

Keywords | Noncommutative Gröbner basis D-A rings Zero divisor Standard representations A-polynomial Overlaprelations AS-reduced |

Length | 20 |

ISSN | 2194-1009 |

URL |
http://www.springer.com/series/10533 |

Language | English |

Journal | Non-Associative and Non-Commutative Algebra and Operator Theory |

Volume | 160 |

Chapter | 9 |

Pages | 137-158 |

Publisher | Springer Proceedings in Mathematics & Statistics |

Address | Dakar |

Year | 2016 |

Edition | 0 |

Translation |
No |

Refereed |
No |

Institution |
Cheikh Anta Diop University |